What is the simplified value of `[ ((1 + x ^(3)) )/( (x ^(2) -1 )) - (( x ^(2) + 1 + x))/( (x + 1)) ] xx (x -1)` ?

Options

A. 1
B. x
C. `2/(x +1)`
D. `1 //(x -1)`

Answer

To simplify the expression \[ \left[ \frac{1 + x^3}{x^2 - 1} - \frac{x^2 + 1 + x}{x + 1} \right] \cdot (x - 1) \], we will follow these steps:

### Step 1: Factor the denominators
The first denominator \(x^2 - 1\) can be factored using the difference of squares:
\[ x^2 - 1 = (x - 1)(x + 1) \]
The second denominator \(x + 1\) is already in its simplest form.

### Step 2: Rewrite the expression
Now we can rewrite the expression:
\[ \left[ \frac{1 + x^3}{(x - 1)(x + 1)} - \frac{x^2 + 1 + x}{x + 1} \right] \cdot (x - 1) \]

### Step 3: Simplify the second term
To combine the fractions, we need a common denominator. The common denominator will be \((x - 1)(x + 1)\):
\[ \frac{x^2 + 1 + x}{x + 1} = \frac{(x^2 + 1 + x)(x - 1)}{(x + 1)(x - 1)} \]
Now we can rewrite the expression:
\[ \frac{1 + x^3 - (x^2 + 1 + x)(x - 1)}{(x - 1)(x + 1)} \cdot (x - 1) \]

### Step 4: Multiply by \((x - 1)\)
When we multiply by \((x - 1)\), we can cancel \((x - 1)\) in the numerator and denominator:
\[ \frac{1 + x^3 - (x^2 + 1 + x)(x - 1)}{x + 1} \]

### Step 5: Expand the numerator
Now we need to expand the numerator:
\[ 1 + x^3 - (x^2 + 1 + x)(x - 1) = 1 + x^3 - (x^3 - x^2 + x - 1) \]
Simplifying this gives:
\[ 1 + x^3 - x^3 + x^2 - x + 1 = x^2 - x + 2 \]

### Step 6: Final expression
Now we have:
\[ \frac{x^2 - x + 2}{x + 1} \]

### Step 7: Conclusion
This is the simplified expression.

### Final Answer:
\[ \frac{x^2 - x + 2}{x + 1} \]

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